How to Find a Percent of a Number

Part = percent × whole. The word "of" between a percent and a number means multiply. A jacket at $25 %$ off, a tip at $18 %$ , a poll where $60 %$ agree, all of them come down to that one multiplication.

Problem

A wildlife reserve counts

200

nesting birds, and

25 %

are herons. How many herons are there?

Show a hint

The word "of" means multiply. Rewrite $25 %$ as $\frac{1}{4}$ or $0.25$ , then multiply it by $200$ .
One quarter of $200$ is $200 \div 4 = 50$ . Or $0.25 \times 200 = 50$ .

Show the full solution

25 % = \frac{1}{4}

, and "of" means multiply, so

25 % of 200 = \frac{1}{4} \times 200 = 50.

There are

50

herons. Decimals work just as well,

0.25 \times 200 = 50

Problem

A bakery sold

35

loaves, and

60 %

were sourdough. How many sourdough loaves were sold?

Show a hint

Rewrite $60 %$ as $0.6$ (or $\frac{3}{5}$ ), then multiply by $35$ .
$0.6 \times 35 = 21$ . Or $\frac{3}{5} \times 35 = 3 \times 7 = 21$ .

Show the full solution

Convert and multiply. Since

60 % = \frac{3}{5} = 0.6

60 % of 35 = \frac{3}{5} \times 35 = 3 \times 7 = 21.

21

sourdough loaves were sold.

Problem

A startup had

15

employees and grew to

300 %

of that size. How many employees are there now?

Show a hint

A percent over $100$ becomes a number bigger than $1$ . Here $300 % = 3$ .
Multiply $3 \times 15 = 45$ .

Show the full solution

300 % = \frac{300}{100} = 3

, so

300 % of 15 = 3 \times 15 = 45.

There are

45

employees. A percent above

100

gives more than the whole, so an answer larger than

15

is the right sign.

Finding

25 %

200

. Split the whole bar of

200

into four equal quarters of

50

. One quarter is

25 %

, so

25 %

200

is one of those quarters,

\frac{1}{4} \times 200 = 50

. Taking a percent of a number is multiplying the whole by that fraction.

Problem

A theater holds

64

seats, and

12 \frac{1}{2} %

are reserved for staff. Since

12 \frac{1}{2} % = \frac{1}{8}

, how many seats are reserved?

Show a hint

First convert. $12 \frac{1}{2} % = 12.5 % = \frac{12.5}{100} = \frac{1}{8}$ .
One eighth of $64$ is $64 \div 8 = 8$ .

Show the full solution

Convert the fractional percent to a clean fraction,

12 \frac{1}{2} % = \frac{12.5}{100} = \frac{1}{8}

. Then

12 \frac{1}{2} % of 64 = \frac{1}{8} \times 64 = 8.

8

seats are reserved.

Problem

A water sample weighs

80

g, and

2.5 %

is dissolved minerals. How many grams of minerals are there?

Show a hint

Slide the point two places left, so $2.5 % = 0.025$ .
$0.025 \times 80 = 2$ .

Show the full solution

Convert the percent to a decimal by sliding the point two places left,

2.5 % = 0.025

. Then

2.5 % of 80 = 0.025 \times 80 = 2.

So there are

2

grams of dissolved minerals.

Problem

A class read

30

books out of a goal of

120

. What percent of the goal did they reach? Give just the number.

Show a hint

Make the fraction part over whole, $\frac{30}{120}$ , and simplify.
$\frac{30}{120} = \frac{1}{4} = 0.25$ , and $0.25$ as a percent is $25 %$ .

Show the full solution

Divide the part by the whole, then write it as a percent.

\frac{30}{120} = \frac{1}{4} = 0.25 = 25 % .

30

25

percent of

120

Problem

A phone battery holds

80

units full and

60

units now. What percent of

80

60

? Give just the number.

Show a hint

The part is $60$ and the whole is $80$ , so form $\frac{60}{80}$ and simplify.
$\frac{60}{80} = \frac{3}{4} = 0.75 = 75 %$ .

Show the full solution

Divide the part by the whole, then convert to a percent.

\frac{60}{80} = \frac{3}{4} = 0.75 = 75 % .

60

75

percent of

80

Problem

A shop bought a desk for

$ 17

and resold it for

$ 51

. What percent is

51

17

? Give just the number.

Show a hint

Form $\frac{51}{17}$ . Since $51$ is bigger than $17$ , expect a percent above $100$ .
$\frac{51}{17} = 3 = 300 %$ .

Show the full solution

Divide the part by the whole,

\frac{51}{17} = 3

, and

3 = 300 %

. So

51

300

percent of

17

. The resale price is three times the cost, which is what a percent above

100

always means.

The three percent questions are one relationship,

part = percent \times whole

, read from three corners. Cover the part and you multiply percent by whole. Cover the percent and you divide the part by the whole. Cover the whole and you divide the part by the percent.

Problem

A scholarship covers

$ 80

of a fee, and that is exactly

20 %

of the full fee.

80

20 %

of what number?

Show a hint

The relationship is part equals percent times whole, so whole equals part divided by percent. Use $20 % = 0.2$ .
$80 \div 0.2 = 400$ . Check, $20 %$ of $400$ is $80$ .

Show the full solution

With

20 % = 0.2

, divide the part by the percent,

whole = \frac{80}{0.2} = 400.

80

20 %

400

. Check by going forward,

0.2 \times 400 = 80

Problem

A hiker has walked

27

km, which is

75 %

of the full trail.

27

75 %

of what number?

Show a hint

Whole equals part divided by percent, so compute $27 \div 0.75$ .
$27 \div \frac{3}{4} = 27 \times \frac{4}{3} = 36$ . Check, $75 %$ of $36$ is $27$ .

Show the full solution

Divide the part by the percent. With

75 % = \frac{3}{4}

whole = 27 \div \frac{3}{4} = 27 \times \frac{4}{3} = 36.

The trail is

36

kilometers long. Check by going forward,

\frac{3}{4} \times 36 = 27

Problem

A startup's revenue this month is

$ 18

k, which is

150 %

of last month's.

18

150 %

of what number?

Show a hint

Whole equals part divided by percent, so compute $18 \div 1.5$ .
$18 \div 1.5 = 12$ . Since the percent is over $100$ , the whole is smaller than the part.

Show the full solution

Divide the part by the percent, with

150 % = 1.5

whole = \frac{18}{1.5} = 12.

Last month's revenue was

12

thousand dollars. Since

150 %

is more than

100 %

, the whole comes out smaller than the part, and

12

is indeed under

18

Problem

A museum finds

30 %

80

visitors came by bus. What percent of

120

school visitors is that same bus count? Give just the number.

Show a hint

Step one, find $30 %$ of $80$ . That is $0.3 \times 80 = 24$ .
Step two, what percent is $24$ of $120$ ? Form $\frac{24}{120} = 0.2 = 20 %$ .

Show the full solution

Do it in two steps. First the part,

30 %

80 = 0.3 \times 80 = 24

. Then ask what percent

24

is of

120

\frac{24}{120} = \frac{1}{5} = 0.2 = 20 % .

30 %

80

20

percent of

120

Practice these ideas

Practice

A farmers market had

90

stalls, and

40 %

sold vegetables. How many stalls sold vegetables?

Show the solution

40 % = 0.4

, so

40 % of 90 = 0.4 \times 90 = 36

. So

36

stalls sold vegetables.

Practice

A song had

12

thousand listeners last week and climbed to

250 %

of that this week. How many thousand listeners this week?

Show the solution

250 % = \frac{250}{100} = 2.5

, so

250 % of 12 = 2.5 \times 12 = 30

. So there are

30

thousand listeners.

Practice

A puzzle has

16

pieces, and

37 \frac{1}{2} %

are edge pieces. How many edge pieces are there?

Show the solution

37 \frac{1}{2} % = \frac{37.5}{100} = \frac{3}{8}

, so

\frac{3}{8} \times 16 = 6

. So there are

6

edge pieces.

Practice

A reservoir holds

150

million liters, and

8 %

is in reserve. How many million liters are in reserve?

Show the solution

8 % = 0.08

, so

8 % of 150 = 0.08 \times 150 = 12

. So

12

million liters are in reserve.

Practice

A team won

18

of its

72

games. What percent did the team win? Give just the number.

Show the solution

\frac{18}{72} = \frac{1}{4} = 0.25 = 25 %

. So the team won

25

percent of its games.

Practice

A jar holds

40

marbles and

30

are blue. What percent of

40

30

? Give just the number.

Show the solution

\frac{30}{40} = \frac{3}{4} = 0.75 = 75 %

. So

30

75

percent of

40

Practice

A sequel sold

84

thousand copies. The original sold

56

thousand. What percent is

84

56

? Give just the number.

Show the solution

\frac{84}{56} = \frac{3}{2} = 1.5 = 150 %

. So

84

150

percent of

56

Practice

A deposit of

$ 21

30 %

of a bike's total price.

21

30 %

of what number?

Show the solution

With

30 % = 0.3

, divide the part by the percent,

\frac{21}{0.3} = 70

. The price is

70

dollars. Check by going forward,

0.3 \times 70 = 21

Practice

A runner has finished

45

laps, which is

90 %

of the race.

45

90 %

of what number?

Show the solution

With

90 % = 0.9

, divide the part by the percent,

\frac{45}{0.9} = 50

. The race is

50

laps. Check by going forward,

0.9 \times 50 = 45

Practice

This year a town recorded

120

building permits, which is

150 %

of last year's.

120

150 %

of what number?

Show the solution

With

150 % = 1.5

, divide the part by the percent,

\frac{120}{1.5} = 80

. Last year had

80

permits. Check by going forward,

1.5 \times 80 = 120

Practice

A tasting portion is

5

g, which is

12 \frac{1}{2} %

of a full serving.

5

12 \frac{1}{2} %

of what number?

Show the solution

12 \frac{1}{2} % = \frac{1}{8}

, and the whole is the part divided by the percent, so

5 \div \frac{1}{8} = 5 \times 8 = 40

. A full serving is

40

grams. Check by going forward,

\frac{1}{8} \times 40 = 5

Practice

Chain two steps. Take

60 %

500

, then take

20 %

of that result. What is

20 %

60 %

500

Show the solution

Work from the inside out. First

60 %

500 = 0.6 \times 500 = 300

. Then

20 %

300 = 0.2 \times 300 = 60

. So

20 %

60 %

500

60

A Percent of a Number

Practice these ideas